Graham's number is celebrated as the largest number ever used in a mathematical proof, although much larger numbers have since claimed this title (such as TREE(3) and SCG(13)). The smallest Bowersism exceeding Graham's number is corporal, and the smallest Saibianism exceeding Graham's number is graatagold.

Contents

Graham's number arose out of the following unsolved problem in Ramsey theory:

Let N* be the smallest dimension n of a hypercube such that if the lines joining all pairs of corners are two-colored for any n ≥ N*, a complete graph K4 of one color with coplanar vertices will be forced. Find N*.

An example of a cube with 12 planar K4's, with a single monochromatic K4 shown below. If you change the edge on the bottom of this K4 to blue, then the cube will contain no monochromatic planar K4's, thus showing that N* is at least 4.

To understand what this problem asks, first consider a hypercube of any number of dimensions (1 dimension would be a line, 2 would be a square, 3 would be a cube, 4 would be a tesseract (4-dimensional cube), etc), and call that number of dimensions N. Then, imagine connecting all possible vertex pairs with lines, and coloring each of those red or blue - one such way you can color all of the vertex pairs of the 3-dimensional cube is shown to the right. What is the smallest number of dimensions N such that all possible colorings would have a monochromatic complete graph of four coplanar vertices (that is a set of four points that are connected in all possible ways, with all lines being the same color)?

Graham published a paper in 1971 proving that the answer exists, providing the upper bound \(F^{7}(12)\), where \(F(n) = 2 \uparrow^{n} 3\) in arrow notation.[2]Sbiis Saibian calls this number "Little Graham". Martin Gardner, when discovering the number's size, found it difficult to explain, and he devised a larger, easier-to-explain number which Graham proved in an unpublished 1977 paper. Martin Gardner wrote about the number in Scientific American and it even made it to Guiness World Records in 1980 as the largest number used in a mathematical proof, although a few years later the title was removed from Guiness World Records.

In 2013, the upper bound was further reduced to N' = 2↑↑2↑↑2↑↑9 using the Hales–Jewett theorem.[3] As of 2014, the best known lower bound for N* is 13.

Since g0 is 4 and not 3, Graham's number cannot be expressed efficiently with most major googological functions. It can be approximated with \(3 → 3 → 64 → 2\) in Conway's chained arrow notation or \(\{3,65,1,2\}\) in BEAF, with upper bound \(\{3, 66, 1, 2\}\). A rare example of an exact representation is Jonathan Bowers' G functions, where it is G644 in base 3. It can be also exactly expressed in the Graham Array Notation as \(\ [[3],4,64]\).

Tim Chow proved that Graham's number is much larger than the Moser.[4] The proof hinges on the fact that, using Steinhaus-Moser Notation, n in a (k + 2)-gon is less than \(n\underbrace{\uparrow\uparrow\ldots\uparrow\uparrow}_{2k-1}n\). He sent the proof to Susan Stepney on July 7, 1998.[5] Coincidentally, Stepney was sent a similar proof by Todd Cesere several days later.

It has been proven that Graham's number is much less than \(\Sigma(64)\),[6] and later a better upper bound \(\Sigma(22)\) was proven.

In the fast-growing hierarchy, Graham's Number can be shown to be less than .

The final digits of Graham's number can be computed by taking advantage of the convergence of last digits, because Graham's number is a power tower of threes. Here is a simple algorithm to obtain the last \(x\) digits \(N(x)\) of Graham's number: